r^2+9r+19=0

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Solution for r^2+9r+19=0 equation: 


Simplifying
r2 + 9r + 19 = 0

Reorder the terms:
19 + 9r + r2 = 0

Solving
19 + 9r + r2 = 0

Solving for variable 'r'.

Begin completing the square.

Move the constant term to the right:

Add '-19' to each side of the equation.
19 + 9r + -19 + r2 = 0 + -19

Reorder the terms:
19 + -19 + 9r + r2 = 0 + -19

Combine like terms: 19 + -19 = 0
0 + 9r + r2 = 0 + -19
9r + r2 = 0 + -19

Combine like terms: 0 + -19 = -19
9r + r2 = -19

The r term is 9r.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9r + 20.25 + r2 = -19 + 20.25

Reorder the terms:
20.25 + 9r + r2 = -19 + 20.25

Combine like terms: -19 + 20.25 = 1.25
20.25 + 9r + r2 = 1.25

Factor a perfect square on the left side:
(r + 4.5)(r + 4.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(r + 4.5) equal to 1.118033989 and -1.118033989.


Subproblem 1
r + 4.5 = 1.118033989

Simplifying
r + 4.5 = 1.118033989

Reorder the terms:
4.5 + r = 1.118033989

Solving
4.5 + r = 1.118033989

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '-4.5' to each side of the equation.
4.5 + -4.5 + r = 1.118033989 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + r = 1.118033989 + -4.5
r = 1.118033989 + -4.5

Combine like terms: 1.118033989 + -4.5 = -3.381966011
r = -3.381966011

Simplifying
r = -3.381966011


Subproblem 2
r + 4.5 = -1.118033989

Simplifying
r + 4.5 = -1.118033989

Reorder the terms:
4.5 + r = -1.118033989

Solving
4.5 + r = -1.118033989

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '-4.5' to each side of the equation.
4.5 + -4.5 + r = -1.118033989 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + r = -1.118033989 + -4.5
r = -1.118033989 + -4.5

Combine like terms: -1.118033989 + -4.5 = -5.618033989
r = -5.618033989

Simplifying
r = -5.618033989


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {-3.381966011, -5.618033989}

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